Most of our analyses are applicable to general fitness and mutation landscapes. However, we characterize in detail the features of artificial selection to direct evolution on quadratic fitness and mutation landscapes, where phenotypes evolve by natural selection towards an evolutionary optimum [1]}. In this case, the impacts of selection and mutation follow linear functions in the high-dimensional phenotypic space, \(\nabla F = - 2 C_0 \cdot {\bf x}\) , \(\nabla M= - 2 L \cdot {\bf x}\) , where \({\bf x}\)  denotes the shifted phenotype vector centered around its stationary state and \(C_0\)  and \(L\)  are selection and mutation matrices, respectively— \(L\)  is a generalization of per-locus mutation rate in high dimensions. The evolutionary model in this case assumes a linear genotype-phenotype map (i.e., \(r_\text{max} =1 \)  in eq. REF ) and a non-linear quadratic phenotype-fitness map, which resembles biophysical models of global epistasis [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}. We can formulate the evolution of mean phenotypes by,
\(d{\bf x}= -2K C\, {\bf x}\, {\text{d}}t + \Sigma \, {\text{d}}{\bf W}\) 
